-
Notifications
You must be signed in to change notification settings - Fork 0
/
IndPrinciples.html
1219 lines (911 loc) · 99.9 KB
/
IndPrinciples.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8"/>
<link href="coqdoc.css" rel="stylesheet" type="text/css"/>
<title>IndPrinciples: Induction Principles</title>
<script type="text/javascript" src="jquery-1.8.3.js"></script>
<script type="text/javascript" src="main.js"></script>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">IndPrinciples<span class="subtitle">Induction Principles</span></h1>
<div class="code code-tight">
</div>
<div class="doc">
<div class="paragraph"> </div>
With the Curry-Howard correspondence and its realization in Coq in
mind, we can now take a deeper look at induction principles.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Require</span> <span class="id" type="keyword">Export</span> <a class="idref" href="ProofObjects.html#"><span class="id" type="library">ProofObjects</span></a>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab259"></a><h1 class="section">Basics</h1>
<div class="paragraph"> </div>
Every time we declare a new <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> datatype, Coq
automatically generates an <i>induction principle</i> for this type.
This induction principle is a theorem like any other: If <span class="inlinecode"><span class="id" type="var">t</span></span> is
defined inductively, the corresponding induction principle is
called <span class="inlinecode"><span class="id" type="var">t_ind</span></span>. Here is the one for natural numbers:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat_ind"><span class="id" type="definition">nat_ind</span></a>.<br/>
<span class="comment">(* ===> nat_ind :<br/>
forall P : nat -> Prop,<br/>
P 0 -><br/>
(forall n : nat, P n -> P (S n)) -><br/>
forall n : nat, P n *)</span><br/>
<br/>
</div>
<div class="doc">
The <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic is a straightforward wrapper that, at its
core, simply performs <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">t_ind</span></span>. To see this more clearly,
let's experiment with directly using <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">nat_ind</span></span>, instead of
the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic, to carry out some proofs. Here, for
example, is an alternate proof of a theorem that we saw in the
<a href="Basics.html"><span class="inlineref">Basics</span></a> chapter.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_0_r'"><span class="id" type="lemma">mult_0_r'</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> * 0 = 0.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat_ind"><span class="id" type="definition">nat_ind</span></a>.<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">intros</span> <span class="id" type="var">n'</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>.<br/>
<span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
This proof is basically the same as the earlier one, but a
few minor differences are worth noting.
<div class="paragraph"> </div>
First, in the induction step of the proof (the <span class="inlinecode">"<span class="id" type="var">S</span>"</span> case), we
have to do a little bookkeeping manually (the <span class="inlinecode"><span class="id" type="tactic">intros</span></span>) that
<span class="inlinecode"><span class="id" type="tactic">induction</span></span> does automatically.
<div class="paragraph"> </div>
Second, we do not introduce <span class="inlinecode"><span class="id" type="var">n</span></span> into the context before applying
<span class="inlinecode"><span class="id" type="var">nat_ind</span></span> — the conclusion of <span class="inlinecode"><span class="id" type="var">nat_ind</span></span> is a quantified formula,
and <span class="inlinecode"><span class="id" type="tactic">apply</span></span> needs this conclusion to exactly match the shape of
the goal state, including the quantifier. By contrast, the
<span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic works either with a variable in the context or
a quantified variable in the goal.
<div class="paragraph"> </div>
These conveniences make <span class="inlinecode"><span class="id" type="tactic">induction</span></span> nicer to use in practice than
applying induction principles like <span class="inlinecode"><span class="id" type="var">nat_ind</span></span> directly. But it is
important to realize that, modulo these bits of bookkeeping,
applying <span class="inlinecode"><span class="id" type="var">nat_ind</span></span> is what we are really doing.
<div class="paragraph"> </div>
<a name="lab260"></a><h4 class="section">Exercise: 2 stars, optional (plus_one_r')</h4>
Complete this proof without using the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_one_r'"><span class="id" type="lemma">plus_one_r'</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> + 1 = <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* FILL IN HERE *)</span> <span class="id" type="var">Admitted</span>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
Coq generates induction principles for every datatype defined with
<span class="inlinecode"><span class="id" type="keyword">Inductive</span></span>, including those that aren't recursive. Although of
course we don't need induction to prove properties of
non-recursive datatypes, the idea of an induction principle still
makes sense for them: it gives a way to prove that a property
holds for all values of the type.
<div class="paragraph"> </div>
These generated principles follow a similar pattern. If we define
a type <span class="inlinecode"><span class="id" type="var">t</span></span> with constructors <span class="inlinecode"><span class="id" type="var">c<sub>1</sub></span></span> ... <span class="inlinecode"><span class="id" type="var">cn</span></span>, Coq generates a
theorem with this shape:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">t_ind</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> : <span class="id" type="var">t</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>,<br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">c<sub>1</sub></span> ... <span style="font-family: arial;">→</span><br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">c<sub>2</sub></span> ... <span style="font-family: arial;">→</span> ...<br/>
... <span class="id" type="tactic">case</span> <span class="id" type="keyword">for</span> <span class="id" type="var">cn</span> ... <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">t</span>, <span class="id" type="var">P</span> <span class="id" type="var">n</span>
<div class="paragraph"> </div>
</div>
The specific shape of each case depends on the arguments to the
corresponding constructor. Before trying to write down a general
rule, let's look at some more examples. First, an example where
the constructors take no arguments:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="yesno"><span class="id" type="inductive">yesno</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="yes"><span class="id" type="constructor">yes</span></a> : <a class="idref" href="IndPrinciples.html#yesno"><span class="id" type="inductive">yesno</span></a><br/>
| <a name="no"><span class="id" type="constructor">no</span></a> : <a class="idref" href="IndPrinciples.html#yesno"><span class="id" type="inductive">yesno</span></a>.<br/>
<br/>
<span class="id" type="keyword">Check</span> <a class="idref" href="IndPrinciples.html#yesno_ind"><span class="id" type="definition">yesno_ind</span></a>.<br/>
<span class="comment">(* ===> yesno_ind : forall P : yesno -> Prop,<br/>
P yes -><br/>
P no -><br/>
forall y : yesno, P y *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab261"></a><h4 class="section">Exercise: 1 star, optional (rgb)</h4>
Write out the induction principle that Coq will generate for the
following datatype. Write down your answer on paper or type it
into a comment, and then compare it with what Coq prints.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="rgb"><span class="id" type="inductive">rgb</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="red"><span class="id" type="constructor">red</span></a> : <a class="idref" href="IndPrinciples.html#rgb"><span class="id" type="inductive">rgb</span></a><br/>
| <a name="green"><span class="id" type="constructor">green</span></a> : <a class="idref" href="IndPrinciples.html#rgb"><span class="id" type="inductive">rgb</span></a><br/>
| <a name="blue"><span class="id" type="constructor">blue</span></a> : <a class="idref" href="IndPrinciples.html#rgb"><span class="id" type="inductive">rgb</span></a>.<br/>
<span class="id" type="keyword">Check</span> <a class="idref" href="IndPrinciples.html#rgb_ind"><span class="id" type="definition">rgb_ind</span></a>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
Here's another example, this time with one of the constructors
taking some arguments.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="natlist"><span class="id" type="inductive">natlist</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="nnil"><span class="id" type="constructor">nnil</span></a> : <a class="idref" href="IndPrinciples.html#natlist"><span class="id" type="inductive">natlist</span></a><br/>
| <a name="ncons"><span class="id" type="constructor">ncons</span></a> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#natlist"><span class="id" type="inductive">natlist</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#natlist"><span class="id" type="inductive">natlist</span></a>.<br/>
<br/>
<span class="id" type="keyword">Check</span> <a class="idref" href="IndPrinciples.html#natlist_ind"><span class="id" type="definition">natlist_ind</span></a>.<br/>
<span class="comment">(* ===> (modulo a little variable renaming)<br/>
natlist_ind :<br/>
forall P : natlist -> Prop,<br/>
P nnil -><br/>
(forall (n : nat) (l : natlist),<br/>
P l -> P (ncons n l)) -><br/>
forall n : natlist, P n *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab262"></a><h4 class="section">Exercise: 1 star, optional (natlist1)</h4>
Suppose we had written the above definition a little
differently:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="natlist1"><span class="id" type="inductive">natlist1</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="nnil1"><span class="id" type="constructor">nnil1</span></a> : <a class="idref" href="IndPrinciples.html#natlist1"><span class="id" type="inductive">natlist1</span></a><br/>
| <a name="nsnoc1"><span class="id" type="constructor">nsnoc1</span></a> : <a class="idref" href="IndPrinciples.html#natlist1"><span class="id" type="inductive">natlist1</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#natlist1"><span class="id" type="inductive">natlist1</span></a>.<br/>
<br/>
</div>
<div class="doc">
Now what will the induction principle look like? <font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
From these examples, we can extract this general rule:
<div class="paragraph"> </div>
<ul class="doclist">
<li> The type declaration gives several constructors; each
corresponds to one clause of the induction principle.
</li>
<li> Each constructor <span class="inlinecode"><span class="id" type="var">c</span></span> takes argument types <span class="inlinecode"><span class="id" type="var">a<sub>1</sub></span></span> ... <span class="inlinecode"><span class="id" type="var">an</span></span>.
</li>
<li> Each <span class="inlinecode"><span class="id" type="var">ai</span></span> can be either <span class="inlinecode"><span class="id" type="var">t</span></span> (the datatype we are defining) or
some other type <span class="inlinecode"><span class="id" type="var">s</span></span>.
</li>
<li> The corresponding case of the induction principle says:
<div class="paragraph"> </div>
<ul class="doclist">
<li> "For all values <span class="inlinecode"><span class="id" type="var">x<sub>1</sub></span></span>...<span class="inlinecode"><span class="id" type="var">xn</span></span> of types <span class="inlinecode"><span class="id" type="var">a<sub>1</sub></span></span>...<span class="inlinecode"><span class="id" type="var">an</span></span>, if <span class="inlinecode"><span class="id" type="var">P</span></span>
holds for each of the inductive arguments (each <span class="inlinecode"><span class="id" type="var">xi</span></span> of type
<span class="inlinecode"><span class="id" type="var">t</span></span>), then <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode"><span class="id" type="var">c</span></span> <span class="inlinecode"><span class="id" type="var">x<sub>1</sub></span></span> <span class="inlinecode">...</span> <span class="inlinecode"><span class="id" type="var">xn</span></span>".
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
<a name="lab263"></a><h4 class="section">Exercise: 1 star, optional (byntree_ind)</h4>
Write out the induction principle that Coq will generate for the
following datatype. (Again, write down your answer on paper or
type it into a comment, and then compare it with what Coq
prints.)
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="byntree"><span class="id" type="inductive">byntree</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="bempty"><span class="id" type="constructor">bempty</span></a> : <a class="idref" href="IndPrinciples.html#byntree"><span class="id" type="inductive">byntree</span></a><br/>
| <a name="bleaf"><span class="id" type="constructor">bleaf</span></a> : <a class="idref" href="IndPrinciples.html#yesno"><span class="id" type="inductive">yesno</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#byntree"><span class="id" type="inductive">byntree</span></a><br/>
| <a name="nbranch"><span class="id" type="constructor">nbranch</span></a> : <a class="idref" href="IndPrinciples.html#yesno"><span class="id" type="inductive">yesno</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#byntree"><span class="id" type="inductive">byntree</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#byntree"><span class="id" type="inductive">byntree</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#byntree"><span class="id" type="inductive">byntree</span></a>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab264"></a><h4 class="section">Exercise: 1 star, optional (ex_set)</h4>
Here is an induction principle for an inductively defined
set.
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">ExSet_ind</span> :<br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">P</span> : <span class="id" type="var">ExSet</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>,<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">b</span> : <span class="id" type="var">bool</span>, <span class="id" type="var">P</span> (<span class="id" type="var">con1</span> <span class="id" type="var">b</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">e</span> : <span class="id" type="var">ExSet</span>), <span class="id" type="var">P</span> <span class="id" type="var">e</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> (<span class="id" type="var">con2</span> <span class="id" type="var">n</span> <span class="id" type="var">e</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">e</span> : <span class="id" type="var">ExSet</span>, <span class="id" type="var">P</span> <span class="id" type="var">e</span>
<div class="paragraph"> </div>
</div>
Give an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> definition of <span class="inlinecode"><span class="id" type="var">ExSet</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="ExSet"><span class="id" type="inductive">ExSet</span></a> : <span class="id" type="keyword">Type</span> :=<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab265"></a><h1 class="section">Polymorphism</h1>
<div class="paragraph"> </div>
Next, what about polymorphic datatypes?
<div class="paragraph"> </div>
The inductive definition of polymorphic lists
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">list</span> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <span class="id" type="var">nil</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span><br/>
| <span class="id" type="var">cons</span> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="var">list</span> <span class="id" type="var">X</span>.
<div class="paragraph"> </div>
</div>
is very similar to that of <span class="inlinecode"><span class="id" type="var">natlist</span></span>. The main difference is
that, here, the whole definition is <i>parameterized</i> on a set <span class="inlinecode"><span class="id" type="var">X</span></span>:
that is, we are defining a <i>family</i> of inductive types <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>,
one for each <span class="inlinecode"><span class="id" type="var">X</span></span>. (Note that, wherever <span class="inlinecode"><span class="id" type="var">list</span></span> appears in the body
of the declaration, it is always applied to the parameter <span class="inlinecode"><span class="id" type="var">X</span></span>.)
The induction principle is likewise parameterized on <span class="inlinecode"><span class="id" type="var">X</span></span>:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">list_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
<span class="id" type="var">P</span> [] <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">x</span> : <span class="id" type="var">X</span>) (<span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>), <span class="id" type="var">P</span> <span class="id" type="var">l</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> (<span class="id" type="var">x</span> :: <span class="id" type="var">l</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>, <span class="id" type="var">P</span> <span class="id" type="var">l</span>
<div class="paragraph"> </div>
</div>
Note that the <i>whole</i> induction principle is parameterized on
<span class="inlinecode"><span class="id" type="var">X</span></span>. That is, <span class="inlinecode"><span class="id" type="var">list_ind</span></span> can be thought of as a polymorphic
function that, when applied to a type <span class="inlinecode"><span class="id" type="var">X</span></span>, gives us back an
induction principle specialized to the type <span class="inlinecode"><span class="id" type="var">list</span></span> <span class="inlinecode"><span class="id" type="var">X</span></span>.
<div class="paragraph"> </div>
<a name="lab266"></a><h4 class="section">Exercise: 1 star, optional (tree)</h4>
Write out the induction principle that Coq will generate for
the following datatype. Compare your answer with what Coq
prints.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="tree"><span class="id" type="inductive">tree</span></a> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="leaf"><span class="id" type="constructor">leaf</span></a> : <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#tree"><span class="id" type="inductive">tree</span></a> <span class="id" type="var">X</span><br/>
| <a name="node"><span class="id" type="constructor">node</span></a> : <a class="idref" href="IndPrinciples.html#tree"><span class="id" type="inductive">tree</span></a> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#tree"><span class="id" type="inductive">tree</span></a> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#tree"><span class="id" type="inductive">tree</span></a> <span class="id" type="var">X</span>.<br/>
<span class="id" type="keyword">Check</span> <a class="idref" href="IndPrinciples.html#tree_ind"><span class="id" type="definition">tree_ind</span></a>.<br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab267"></a><h4 class="section">Exercise: 1 star, optional (mytype)</h4>
Find an inductive definition that gives rise to the
following induction principle:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">mytype_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">mytype</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span> : <span class="id" type="var">X</span>, <span class="id" type="var">P</span> (<span class="id" type="var">constr1</span> <span class="id" type="var">X</span> <span class="id" type="var">x</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">P</span> (<span class="id" type="var">constr2</span> <span class="id" type="var">X</span> <span class="id" type="var">n</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">m</span> : <span class="id" type="var">mytype</span> <span class="id" type="var">X</span>, <span class="id" type="var">P</span> <span class="id" type="var">m</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">P</span> (<span class="id" type="var">constr3</span> <span class="id" type="var">X</span> <span class="id" type="var">m</span> <span class="id" type="var">n</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">m</span> : <span class="id" type="var">mytype</span> <span class="id" type="var">X</span>, <span class="id" type="var">P</span> <span class="id" type="var">m</span>
<div class="paragraph"> </div>
</div>
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab268"></a><h4 class="section">Exercise: 1 star, optional (foo)</h4>
Find an inductive definition that gives rise to the
following induction principle:
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">foo_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> <span class="id" type="var">Y</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">foo</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">x</span> : <span class="id" type="var">X</span>, <span class="id" type="var">P</span> (<span class="id" type="var">bar</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span class="id" type="var">x</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">y</span> : <span class="id" type="var">Y</span>, <span class="id" type="var">P</span> (<span class="id" type="var">baz</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span class="id" type="var">y</span>)) <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">f<sub>1</sub></span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="var">foo</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span>,<br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">P</span> (<span class="id" type="var">f<sub>1</sub></span> <span class="id" type="var">n</span>)) <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> (<span class="id" type="var">quux</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span> <span class="id" type="var">f<sub>1</sub></span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">f<sub>2</sub></span> : <span class="id" type="var">foo</span> <span class="id" type="var">X</span> <span class="id" type="var">Y</span>, <span class="id" type="var">P</span> <span class="id" type="var">f<sub>2</sub></span>
<div class="paragraph"> </div>
</div>
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
<div class="paragraph"> </div>
<a name="lab269"></a><h4 class="section">Exercise: 1 star, optional (foo')</h4>
Consider the following inductive definition:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="foo'"><span class="id" type="inductive">foo'</span></a> (<span class="id" type="var">X</span>:<span class="id" type="keyword">Type</span>) : <span class="id" type="keyword">Type</span> :=<br/>
| <a name="C<sub>1</sub>"><span class="id" type="constructor">C<sub>1</sub></span></a> : <a class="idref" href="Poly.html#list"><span class="id" type="inductive">list</span></a> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#foo'"><span class="id" type="inductive">foo'</span></a> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <a class="idref" href="IndPrinciples.html#foo'"><span class="id" type="inductive">foo'</span></a> <span class="id" type="var">X</span><br/>
| <a name="C<sub>2</sub>"><span class="id" type="constructor">C<sub>2</sub></span></a> : <a class="idref" href="IndPrinciples.html#foo'"><span class="id" type="inductive">foo'</span></a> <span class="id" type="var">X</span>.<br/>
<br/>
</div>
<div class="doc">
What induction principle will Coq generate for <span class="inlinecode"><span class="id" type="var">foo'</span></span>? Fill
in the blanks, then check your answer with Coq.)
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">foo'_ind</span> :<br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">X</span> : <span class="id" type="keyword">Type</span>) (<span class="id" type="var">P</span> : <span class="id" type="var">foo'</span> <span class="id" type="var">X</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">l</span> : <span class="id" type="var">list</span> <span class="id" type="var">X</span>) (<span class="id" type="var">f</span> : <span class="id" type="var">foo'</span> <span class="id" type="var">X</span>),<br/>
<span class="id" type="var">_______________________</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">_______________________</span> ) <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">___________________________________________</span> <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">f</span> : <span class="id" type="var">foo'</span> <span class="id" type="var">X</span>, <span class="id" type="var">________________________</span>
<div class="paragraph"> </div>
</div>
<div class="paragraph"> </div>
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab270"></a><h1 class="section">Induction Hypotheses</h1>
<div class="paragraph"> </div>
Where does the phrase "induction hypothesis" fit into this story?
<div class="paragraph"> </div>
The induction principle for numbers
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="font-family: arial;">∀</span><span class="id" type="var">P</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>,<br/>
<span class="id" type="var">P</span> 0 <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">P</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">P</span> <span class="id" type="var">n</span>
<div class="paragraph"> </div>
</div>
is a generic statement that holds for all propositions
<span class="inlinecode"><span class="id" type="var">P</span></span> (or rather, strictly speaking, for all families of
propositions <span class="inlinecode"><span class="id" type="var">P</span></span> indexed by a number <span class="inlinecode"><span class="id" type="var">n</span></span>). Each time we
use this principle, we are choosing <span class="inlinecode"><span class="id" type="var">P</span></span> to be a particular
expression of type <span class="inlinecode"><span class="id" type="var">nat</span><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span></span>.
<div class="paragraph"> </div>
We can make proofs by induction more explicit by giving
this expression a name. For example, instead of stating
the theorem <span class="inlinecode"><span class="id" type="var">mult_0_r</span></span> as "<span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n</span>,</span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">*</span> <span class="inlinecode">0</span> <span class="inlinecode">=</span> <span class="inlinecode">0</span>," we can
write it as "<span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n</span>,</span> <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>", where <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> is defined
as...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <a name="P_m0r"><span class="id" type="definition">P_m0r</span></a> (<span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) : <span class="id" type="keyword">Prop</span> :=<br/>
<a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> * 0 = 0.<br/>
<br/>
</div>
<div class="doc">
... or equivalently:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Definition</span> <a name="P_m0r'"><span class="id" type="definition">P_m0r'</span></a> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a><span style="font-family: arial;">→</span><span class="id" type="keyword">Prop</span> :=<br/>
<span class="id" type="keyword">fun</span> <span class="id" type="var">n</span> ⇒ <a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> * 0 = 0.<br/>
<br/>
</div>
<div class="doc">
Now it is easier to see where <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> appears in the proof.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="mult_0_r''"><span class="id" type="lemma">mult_0_r''</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#P_m0r"><span class="id" type="definition">P_m0r</span></a> <a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat_ind"><span class="id" type="definition">nat_ind</span></a>.<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="comment">(* Note the proof state at this point! *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">IHn</span>.<br/>
<span class="id" type="tactic">unfold</span> <a class="idref" href="IndPrinciples.html#P_m0r"><span class="id" type="definition">P_m0r</span></a> <span class="id" type="keyword">in</span> <span class="id" type="var">IHn</span>. <span class="id" type="tactic">unfold</span> <a class="idref" href="IndPrinciples.html#P_m0r"><span class="id" type="definition">P_m0r</span></a>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">apply</span> <span class="id" type="var">IHn</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
This extra naming step isn't something that we do in
normal proofs, but it is useful to do it explicitly for an example
or two, because it allows us to see exactly what the induction
hypothesis is. If we prove <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n</span>,</span> <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> by induction on
<span class="inlinecode"><span class="id" type="var">n</span></span> (using either <span class="inlinecode"><span class="id" type="tactic">induction</span></span> or <span class="inlinecode"><span class="id" type="tactic">apply</span></span> <span class="inlinecode"><span class="id" type="var">nat_ind</span></span>), we see that the
first subgoal requires us to prove <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode">0</span> ("<span class="inlinecode"><span class="id" type="var">P</span></span> holds for
zero"), while the second subgoal requires us to prove <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>,</span>
<span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> <span class="inlinecode"><span style="font-family: arial;">→</span></span> <span class="inlinecode"><span class="id" type="var">P_m0r</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span> (that is "<span class="inlinecode"><span class="id" type="var">P</span></span> holds of <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> if it
holds of <span class="inlinecode"><span class="id" type="var">n'</span></span>" or, more elegantly, "<span class="inlinecode"><span class="id" type="var">P</span></span> is preserved by <span class="inlinecode"><span class="id" type="var">S</span></span>").
The <i>induction hypothesis</i> is the premise of this latter
implication — the assumption that <span class="inlinecode"><span class="id" type="var">P</span></span> holds of <span class="inlinecode"><span class="id" type="var">n'</span></span>, which we are
allowed to use in proving that <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span>.
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab271"></a><h1 class="section">More on the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> Tactic</h1>
<div class="paragraph"> </div>
The <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic actually does even more low-level
bookkeeping for us than we discussed above.
<div class="paragraph"> </div>
Recall the informal statement of the induction principle for
natural numbers:
<div class="paragraph"> </div>
<ul class="doclist">
<li> If <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> is some proposition involving a natural number n, and
we want to show that P holds for <i>all</i> numbers n, we can
reason like this:
<ul class="doclist">
<li> show that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">O</span></span> holds
</li>
<li> show that, if <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> holds, then so does <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span>
</li>
<li> conclude that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> holds for all n.
</li>
</ul>
</li>
</ul>
So, when we begin a proof with <span class="inlinecode"><span class="id" type="tactic">intros</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> and then <span class="inlinecode"><span class="id" type="tactic">induction</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span>,
we are first telling Coq to consider a <i>particular</i> <span class="inlinecode"><span class="id" type="var">n</span></span> (by
introducing it into the context) and then telling it to prove
something about <i>all</i> numbers (by using induction).
<div class="paragraph"> </div>
What Coq actually does in this situation, internally, is to
"re-generalize" the variable we perform induction on. For
example, in our original proof that <span class="inlinecode"><span class="id" type="var">plus</span></span> is associative...
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_assoc'"><span class="id" type="lemma">plus_assoc'</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> + (<a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="IndPrinciples.html#p"><span class="id" type="variable">p</span></a>) = (<a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a>) + <a class="idref" href="IndPrinciples.html#p"><span class="id" type="variable">p</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* ...we first introduce all 3 variables into the context,<br/>
which amounts to saying "Consider an arbitrary <span class="inlinecode"><span class="id" type="var">n</span></span>, <span class="inlinecode"><span class="id" type="var">m</span></span>, and<br/>
<span class="inlinecode"><span class="id" type="var">p</span></span>..." *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">n</span> <span class="id" type="var">m</span> <span class="id" type="var">p</span>.<br/>
<span class="comment">(* ...We now use the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic to prove <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> (that<br/>
is, <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode">(<span class="id" type="var">m</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">p</span>)</span> <span class="inlinecode">=</span> <span class="inlinecode">(<span class="id" type="var">n</span></span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">m</span>)</span> <span class="inlinecode">+</span> <span class="inlinecode"><span class="id" type="var">p</span></span>) for _all_ <span class="inlinecode"><span class="id" type="var">n</span></span>,<br/>
and hence also for the particular <span class="inlinecode"><span class="id" type="var">n</span></span> that is in the context<br/>
at the moment. *)</span><br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span><br/>
<span class="comment">(* In the second subgoal generated by <span class="inlinecode"><span class="id" type="tactic">induction</span></span> -- the<br/>
"inductive step" -- we must prove that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> implies<br/>
<span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span> for all <span class="inlinecode"><span class="id" type="var">n'</span></span>. The <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic<br/>
automatically introduces <span class="inlinecode"><span class="id" type="var">n'</span></span> and <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n'</span></span> into the context<br/>
for us, leaving just <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n'</span>)</span> as the goal. *)</span><br/>
<span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
It also works to apply <span class="inlinecode"><span class="id" type="tactic">induction</span></span> to a variable that is
quantified in the goal.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_comm'"><span class="id" type="lemma">plus_comm'</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a> = <a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">n</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">n'</span>].<br/>
- <span class="comment">(* n = O *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="Induction.html#plus_n_O"><span class="id" type="lemma">plus_n_O</span></a>. <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* n = S n' *)</span> <span class="id" type="tactic">intros</span> <span class="id" type="var">m</span>. <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">→</span> <span class="id" type="var">IHn'</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="Induction.html#plus_n_Sm"><span class="id" type="axiom">plus_n_Sm</span></a>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Note that <span class="inlinecode"><span class="id" type="tactic">induction</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> leaves <span class="inlinecode"><span class="id" type="var">m</span></span> still bound in the goal —
i.e., what we are proving inductively is a statement beginning
with <span class="inlinecode"><span style="font-family: arial;">∀</span></span> <span class="inlinecode"><span class="id" type="var">m</span></span>.
<div class="paragraph"> </div>
If we do <span class="inlinecode"><span class="id" type="tactic">induction</span></span> on a variable that is quantified in the goal
<i>after</i> some other quantifiers, the <span class="inlinecode"><span class="id" type="tactic">induction</span></span> tactic will
automatically introduce the variables bound by these quantifiers
into the context.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="plus_comm''"><span class="id" type="lemma">plus_comm''</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> <span class="id" type="var">m</span> : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>,<br/>
<a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> + <a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a> = <a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a> + <a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="comment">(* Let's do induction on <span class="inlinecode"><span class="id" type="var">m</span></span> this time, instead of <span class="inlinecode"><span class="id" type="var">n</span></span>... *)</span><br/>
<span class="id" type="tactic">induction</span> <span class="id" type="var">m</span> <span class="id" type="keyword">as</span> [| <span class="id" type="var">m'</span>].<br/>
- <span class="comment">(* m = O *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="Induction.html#plus_n_O"><span class="id" type="lemma">plus_n_O</span></a>. <span class="id" type="tactic">reflexivity</span>.<br/>
- <span class="comment">(* m = S m' *)</span> <span class="id" type="tactic">simpl</span>. <span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <span class="id" type="var">IHm'</span>.<br/>
<span class="id" type="tactic">rewrite</span> <span style="font-family: arial;">←</span> <a class="idref" href="Induction.html#plus_n_Sm"><span class="id" type="axiom">plus_n_Sm</span></a>. <span class="id" type="tactic">reflexivity</span>. <span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
<a name="lab272"></a><h4 class="section">Exercise: 1 star, optional (plus_explicit_prop)</h4>
Rewrite both <span class="inlinecode"><span class="id" type="var">plus_assoc'</span></span> and <span class="inlinecode"><span class="id" type="var">plus_comm'</span></span> and their proofs in
the same style as <span class="inlinecode"><span class="id" type="var">mult_0_r''</span></span> above — that is, for each theorem,
give an explicit <span class="inlinecode"><span class="id" type="keyword">Definition</span></span> of the proposition being proved by
induction, and state the theorem and proof in terms of this
defined proposition.
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* FILL IN HERE *)</span><br/>
</div>
<div class="doc">
<font size=-2 color="rgb(80%,80%,100%)">☐</font>
</div>
<div class="code code-tight">
<br/>
</div>
<div class="doc">
<a name="lab273"></a><h1 class="section">Induction Principles in <span class="inlinecode"><span class="id" type="keyword">Prop</span></span></h1>
<div class="paragraph"> </div>
Earlier, we looked in detail at the induction principles that Coq
generates for inductively defined <i>sets</i>. The induction
principles for inductively defined <i>propositions</i> like <span class="inlinecode"><span class="id" type="var">ev</span></span> are a
tiny bit more complicated. As with all induction principles, we
want to use the induction principle on <span class="inlinecode"><span class="id" type="var">ev</span></span> to prove things by
inductively considering the possible shapes that something in <span class="inlinecode"><span class="id" type="var">ev</span></span>
can have. Intuitively speaking, however, what we want to prove
are not statements about <i>evidence</i> but statements about
<i>numbers</i>: accordingly, we want an induction principle that lets
us prove properties of numbers by induction on evidence.
<div class="paragraph"> </div>
For example, from what we've said so far, you might expect the
inductive definition of <span class="inlinecode"><span class="id" type="var">ev</span></span>...
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="keyword">Inductive</span> <span class="id" type="var">ev</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <span class="id" type="var">ev_0</span> : <span class="id" type="var">ev</span> 0<br/>
| <span class="id" type="var">ev_SS</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">ev</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">ev</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">n</span>)).
<div class="paragraph"> </div>
</div>
...to give rise to an induction principle that looks like this...
<div class="paragraph"> </div>
<div class="code code-tight">
<span class="id" type="var">ev_ind_max</span> : <span style="font-family: arial;">∀</span><span class="id" type="var">P</span> : (<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>, <span class="id" type="var">ev</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>),<br/>
<span class="id" type="var">P</span> <span class="id" type="var">O</span> <span class="id" type="var">ev_0</span> <span style="font-family: arial;">→</span><br/>
(<span style="font-family: arial;">∀</span>(<span class="id" type="var">m</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">E</span> : <span class="id" type="var">ev</span> <span class="id" type="var">m</span>),<br/>
<span class="id" type="var">P</span> <span class="id" type="var">m</span> <span class="id" type="var">E</span> <span style="font-family: arial;">→</span><br/>
<span class="id" type="var">P</span> (<span class="id" type="var">S</span> (<span class="id" type="var">S</span> <span class="id" type="var">m</span>)) (<span class="id" type="var">ev_SS</span> <span class="id" type="var">m</span> <span class="id" type="var">E</span>)) <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span>(<span class="id" type="var">n</span> : <span class="id" type="var">nat</span>) (<span class="id" type="var">E</span> : <span class="id" type="var">ev</span> <span class="id" type="var">n</span>),<br/>
<span class="id" type="var">P</span> <span class="id" type="var">n</span> <span class="id" type="var">E</span>
<div class="paragraph"> </div>
</div>
... because:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Since <span class="inlinecode"><span class="id" type="var">ev</span></span> is indexed by a number <span class="inlinecode"><span class="id" type="var">n</span></span> (every <span class="inlinecode"><span class="id" type="var">ev</span></span> object <span class="inlinecode"><span class="id" type="var">E</span></span> is
a piece of evidence that some particular number <span class="inlinecode"><span class="id" type="var">n</span></span> is even),
the proposition <span class="inlinecode"><span class="id" type="var">P</span></span> is parameterized by both <span class="inlinecode"><span class="id" type="var">n</span></span> and <span class="inlinecode"><span class="id" type="var">E</span></span> —
that is, the induction principle can be used to prove
assertions involving both an even number and the evidence that
it is even.
<div class="paragraph"> </div>
</li>
<li> Since there are two ways of giving evidence of evenness (<span class="inlinecode"><span class="id" type="var">ev</span></span>
has two constructors), applying the induction principle
generates two subgoals:
<div class="paragraph"> </div>
<ul class="doclist">
<li> We must prove that <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode"><span class="id" type="var">O</span></span> and <span class="inlinecode"><span class="id" type="var">ev_0</span></span>.
<div class="paragraph"> </div>
</li>
<li> We must prove that, whenever <span class="inlinecode"><span class="id" type="var">n</span></span> is an even number and <span class="inlinecode"><span class="id" type="var">E</span></span>
is an evidence of its evenness, if <span class="inlinecode"><span class="id" type="var">P</span></span> holds of <span class="inlinecode"><span class="id" type="var">n</span></span> and
<span class="inlinecode"><span class="id" type="var">E</span></span>, then it also holds of <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span> and <span class="inlinecode"><span class="id" type="var">ev_SS</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> <span class="inlinecode"><span class="id" type="var">E</span></span>.
<div class="paragraph"> </div>
</li>
</ul>
</li>
<li> If these subgoals can be proved, then the induction principle
tells us that <span class="inlinecode"><span class="id" type="var">P</span></span> is true for <i>all</i> even numbers <span class="inlinecode"><span class="id" type="var">n</span></span> and
evidence <span class="inlinecode"><span class="id" type="var">E</span></span> of their evenness.
</li>
</ul>
<div class="paragraph"> </div>
This is more flexibility than we normally need or want: it is
giving us a way to prove logical assertions where the assertion
involves properties of some piece of <i>evidence</i> of evenness, while
all we really care about is proving properties of <i>numbers</i> that
are even — we are interested in assertions about numbers, not
about evidence. It would therefore be more convenient to have an
induction principle for proving propositions <span class="inlinecode"><span class="id" type="var">P</span></span> that are
parameterized just by <span class="inlinecode"><span class="id" type="var">n</span></span> and whose conclusion establishes <span class="inlinecode"><span class="id" type="var">P</span></span> for
all even numbers <span class="inlinecode"><span class="id" type="var">n</span></span>:
<div class="paragraph"> </div>
<div class="code code-tight">
<span style="font-family: arial;">∀</span><span class="id" type="var">P</span> : <span class="id" type="var">nat</span> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span>,<br/>
... <span style="font-family: arial;">→</span><br/>
<span style="font-family: arial;">∀</span><span class="id" type="var">n</span> : <span class="id" type="var">nat</span>,<br/>
<span class="id" type="var">even</span> <span class="id" type="var">n</span> <span style="font-family: arial;">→</span> <span class="id" type="var">P</span> <span class="id" type="var">n</span>
<div class="paragraph"> </div>
</div>
For this reason, Coq actually generates the following simplified
induction principle for <span class="inlinecode"><span class="id" type="var">ev</span></span>:
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <a class="idref" href="IndProp.html#ev_ind"><span class="id" type="definition">ev_ind</span></a>.<br/>
<span class="comment">(* ===> ev_ind<br/>
: forall P : nat -> Prop,<br/>
P 0 -><br/>
(forall n : nat, ev n -> P n -> P (S (S n))) -><br/>
forall n : nat,<br/>
ev n -> P n *)</span><br/>
<br/>
</div>
<div class="doc">
In particular, Coq has dropped the evidence term <span class="inlinecode"><span class="id" type="var">E</span></span> as a
parameter of the the proposition <span class="inlinecode"><span class="id" type="var">P</span></span>.
<div class="paragraph"> </div>
In English, <span class="inlinecode"><span class="id" type="var">ev_ind</span></span> says:
<div class="paragraph"> </div>
<ul class="doclist">
<li> Suppose, <span class="inlinecode"><span class="id" type="var">P</span></span> is a property of natural numbers (that is, <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> is
a <span class="inlinecode"><span class="id" type="keyword">Prop</span></span> for every <span class="inlinecode"><span class="id" type="var">n</span></span>). To show that <span class="inlinecode"><span class="id" type="var">P</span></span> <span class="inlinecode"><span class="id" type="var">n</span></span> holds whenever <span class="inlinecode"><span class="id" type="var">n</span></span>
is even, it suffices to show:
<div class="paragraph"> </div>
<ul class="doclist">
<li> <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode">0</span>,
<div class="paragraph"> </div>
</li>
<li> for any <span class="inlinecode"><span class="id" type="var">n</span></span>, if <span class="inlinecode"><span class="id" type="var">n</span></span> is even and <span class="inlinecode"><span class="id" type="var">P</span></span> holds for <span class="inlinecode"><span class="id" type="var">n</span></span>, then <span class="inlinecode"><span class="id" type="var">P</span></span>
holds for <span class="inlinecode"><span class="id" type="var">S</span></span> <span class="inlinecode">(<span class="id" type="var">S</span></span> <span class="inlinecode"><span class="id" type="var">n</span>)</span>.
</li>
</ul>
</li>
</ul>
<div class="paragraph"> </div>
As expected, we can apply <span class="inlinecode"><span class="id" type="var">ev_ind</span></span> directly instead of using
<span class="inlinecode"><span class="id" type="tactic">induction</span></span>.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Theorem</span> <a name="ev_ev'"><span class="id" type="lemma">ev_ev'</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">n</span>, <a class="idref" href="IndProp.html#ev"><span class="id" type="inductive">ev</span></a> <a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a> <span style="font-family: arial;">→</span> <a class="idref" href="IndProp.html#ev'"><span class="id" type="inductive">ev'</span></a> <a class="idref" href="IndPrinciples.html#n"><span class="id" type="variable">n</span></a>.<br/>
<span class="id" type="keyword">Proof</span>.<br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="IndProp.html#ev_ind"><span class="id" type="definition">ev_ind</span></a>.<br/>
- <span class="comment">(* ev_0 *)</span><br/>
<span class="id" type="tactic">apply</span> <a class="idref" href="IndProp.html#ev'_0"><span class="id" type="constructor">ev'_0</span></a>.<br/>
- <span class="comment">(* ev_SS *)</span><br/>
<span class="id" type="tactic">intros</span> <span class="id" type="var">m</span> <span class="id" type="var">Hm</span> <span class="id" type="var">IH</span>.<br/>
<span class="id" type="tactic">apply</span> (<a class="idref" href="IndProp.html#ev'_sum"><span class="id" type="constructor">ev'_sum</span></a> 2 <span class="id" type="var">m</span>).<br/>
+ <span class="id" type="tactic">apply</span> <a class="idref" href="IndProp.html#ev'_2"><span class="id" type="constructor">ev'_2</span></a>.<br/>
+ <span class="id" type="tactic">apply</span> <span class="id" type="var">IH</span>.<br/>
<span class="id" type="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
The precise form of an <span class="inlinecode"><span class="id" type="keyword">Inductive</span></span> definition can affect the
induction principle Coq generates.
<div class="paragraph"> </div>
For example, in chapter <a href="IndProp.html"><span class="inlineref">IndProp</span></a>, we defined <span class="inlinecode">≤</span> as:
</div>
<div class="code code-tight">
<br/>
<span class="comment">(* Inductive le : nat -> nat -> Prop :=<br/>
| le_n : forall n, le n n<br/>
| le_S : forall n m, (le n m) -> (le n (S m)). *)</span><br/>
<br/>
</div>
<div class="doc">
This definition can be streamlined a little by observing that the
left-hand argument <span class="inlinecode"><span class="id" type="var">n</span></span> is the same everywhere in the definition,
so we can actually make it a "general parameter" to the whole
definition, rather than an argument to each constructor.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Inductive</span> <a name="le"><span class="id" type="inductive">le</span></a> (<span class="id" type="var">n</span>:<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a>) : <a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#nat"><span class="id" type="inductive">nat</span></a> <span style="font-family: arial;">→</span> <span class="id" type="keyword">Prop</span> :=<br/>
| <a name="le_n"><span class="id" type="constructor">le_n</span></a> : <a class="idref" href="IndPrinciples.html#le"><span class="id" type="inductive">le</span></a> <span class="id" type="var">n</span> <span class="id" type="var">n</span><br/>
| <a name="le_S"><span class="id" type="constructor">le_S</span></a> : <span style="font-family: arial;">∀</span><span class="id" type="var">m</span>, (<a class="idref" href="IndPrinciples.html#le"><span class="id" type="inductive">le</span></a> <span class="id" type="var">n</span> <a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a>) <span style="font-family: arial;">→</span> (<a class="idref" href="IndPrinciples.html#le"><span class="id" type="inductive">le</span></a> <span class="id" type="var">n</span> (<a class="idref" href="http://coq.inria.fr/library/Coq.Init.Datatypes.html#S"><span class="id" type="constructor">S</span></a> <a class="idref" href="IndPrinciples.html#m"><span class="id" type="variable">m</span></a>)).<br/>
<br/>
<span class="id" type="keyword">Notation</span> "m ≤ n" := (<a class="idref" href="IndPrinciples.html#le"><span class="id" type="inductive">le</span></a> <span class="id" type="var">m</span> <span class="id" type="var">n</span>).<br/>
<br/>
</div>
<div class="doc">
The second one is better, even though it looks less symmetric.
Why? Because it gives us a simpler induction principle.
</div>
<div class="code code-tight">
<br/>
<span class="id" type="keyword">Check</span> <a class="idref" href="IndPrinciples.html#le_ind"><span class="id" type="definition">le_ind</span></a>.<br/>
<span class="comment">(* ===> forall (n : nat) (P : nat -> Prop),<br/>
P n -><br/>
(forall m : nat, n <= m -> P m -> P (S m)) -><br/>
forall n<sub>0</sub> : nat, n <= n<sub>0</sub> -> P n<sub>0</sub> *)</span><br/>
<br/>
</div>
<div class="doc">
<a name="lab274"></a><h1 class="section">Formal vs. Informal Proofs by Induction</h1>
<div class="paragraph"> </div>
Question: What is the relation between a formal proof of a
proposition <span class="inlinecode"><span class="id" type="var">P</span></span> and an informal proof of the same proposition <span class="inlinecode"><span class="id" type="var">P</span></span>?
<div class="paragraph"> </div>
Answer: The latter should <i>teach</i> the reader how to produce the
former.
<div class="paragraph"> </div>
Question: How much detail is needed??
<div class="paragraph"> </div>
Unfortunately, there is no single right answer; rather, there is a
range of choices.
<div class="paragraph"> </div>
At one end of the spectrum, we can essentially give the reader the
whole formal proof (i.e., the "informal" proof will amount to just
transcribing the formal one into words). This may give the reader
the ability to reproduce the formal one for themselves, but it
probably doesn't <i>teach</i> them anything much.
<div class="paragraph"> </div>
At the other end of the spectrum, we can say "The theorem is true
and you can figure out why for yourself if you think about it hard
enough." This is also not a good teaching strategy, because often
writing the proof requires one or more significant insights into
the thing we're proving, and most readers will give up before they
rediscover all the same insights as we did.
<div class="paragraph"> </div>
In the middle is the golden mean — a proof that includes all of
the essential insights (saving the reader the hard work that we
went through to find the proof in the first place) plus high-level
suggestions for the more routine parts to save the reader from
spending too much time reconstructing these (e.g., what the IH says
and what must be shown in each case of an inductive proof), but not
so much detail that the main ideas are obscured.
<div class="paragraph"> </div>
Since we've spent much of this chapter looking "under the hood" at
formal proofs by induction, now is a good moment to talk a little
about <i>informal</i> proofs by induction.
<div class="paragraph"> </div>
In the real world of mathematical communication, written proofs
range from extremely longwinded and pedantic to extremely brief and
telegraphic. Although the ideal is somewhere in between, while one
is getting used to the style it is better to start out at the
pedantic end. Also, during the learning phase, it is probably
helpful to have a clear standard to compare against. With this in
mind, we offer two templates — one for proofs by induction over
<i>data</i> (i.e., where the thing we're doing induction on lives in
<span class="inlinecode"><span class="id" type="keyword">Type</span></span>) and one for proofs by induction over <i>evidence</i> (i.e.,
where the inductively defined thing lives in <span class="inlinecode"><span class="id" type="keyword">Prop</span></span>).
<div class="paragraph"> </div>
<a name="lab275"></a><h2 class="section">Induction Over an Inductively Defined Set</h2>